Projected Primal-Dual Dynamics for Distributed Constrained Nonsmooth Convex Optimization

• Published in: IEEE transactions on cybernetics Vol. 50; no. 4; pp. 1776 - 1782
• Main Authors: Zhu, Yanan, Yu, Wenwu, Wen, Guanghui, Chen, Guanrong
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.04.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Constraints; Convergence; Convex analysis; Convex functions; Convexity; Cybernetics; Distributed convex optimization; Dynamic stability; Dynamics; Heuristic algorithms; Indexes; Linear programming; multiagent networks; Multiagent systems; nonsmooth analysis; Optimization; primal–dual dynamics; Stability analysis; Topology
• ISSN: 2168-2267; 2168-2275; 2168-2275
• DOI: 10.1109/TCYB.2018.2883095

A distributed nonsmooth convex optimization problem subject to a general type of constraint, including equality and inequality as well as bounded constraints, is studied in this paper for a multiagent network with a fixed and connected communication topology. To collectively solve such a complex optimization problem, primal-dual dynamics with projection operation are investigated under optimal conditions. For the nonsmooth convex optimization problem, a framework under the LaSalle's invariance principle from nonsmooth analysis is established, where the asymptotic stability of the primal-dual dynamics at an optimal solution is guaranteed. For the case where inequality and bounded constraints are not involved and the objective function is twice differentiable and strongly convex, the globally exponential convergence of the primal- dual dynamics is established. Finally, two simulations are provided to verify and visualize the theoretical results.